Do you know if it is common for the bird to get injured in this process?
I don't, though being injured in the process isn't a forgone conclusion. And not everyone just turns a bird loose in the yard and lets the pups group to together and chase willy-nilly as is shown in the video.
Shortly after MacNut came home one of my quail (egg layers, about pigeon-sized) escaped and I didn't realize it. She found it and trotted back up to the house to show hubby while I was working in the barn -- I let her go once I made sure hubby knew she was coming, thinking she just had snuck a toy out. Five or ten minutes later hubby realized she was carrying around a quail when she put it down for a second and it fluttered before she picked it back up and continued walking around.
Despite all that time and at least two pick-ups, the quail didn't even have a bent feather and is to this day laying eggs out in the barn. Naturally soft mouths are cool.
Really wish they would use dead birds.
Many do. It's like any other aspect of puppy testing in the sense that no one really agrees on any one best way of doing it. Some use killed birds due to lack of availability of live birds. Others use killed birds because they don't like using a live bird for sensitivity reasons or because they don't want the pups' first experience to be with something that may fly at them and scare them.
The main goal should be to determine, one way or another, which pups show strong birdiness to help figure out which pups are going to working hunt/field homes. Shot birds don't always die and you want a dog who is going to persist in finding and bringing in a wounded bird rather than let them die on their own out in the field. Which is doubly bad -- not only does that bird suffer more, but another bird may be shot to reach the bag limit in its place.
How that natural proclivity is best determined, I will leave to more knowledgeable people than I.